Can anyone school me on a non-destructive test to determine compressive strength of an existing round columns (test cylinders, of coarse, have been disposed of)?

I recommend that you visit the PCA website and find the reference entitled: Design and Control of Concrete Mixtures.  This booklet is the industry standard on concrete before and after the placing.

http://www.portcement.org/info_resources/

drew...
There is not a definitive nondestructive test for concrete that doesn't require correlation.  If you have several columns to check, you can approach this by the use of limited destructive testing coupled with nondestructive testing.  Here's the approach I would take.....

Drill three, 2-inch diameter cores into one or two adjacent columns.  Have a pulse-velocity measurement taken on these cores, then destructively test them in compression.  The cores should have a finished length of twice the diameter for most relevant results.

Once you have established a correlation between the pulse-velocity measurement and the compressive strength, you can then use the pulse-velocity device to check the other columns with a reasonable degree of accuracy.

Another, but less accurate method is to use a Windsor Probe.  This should be correlated as well.

Don't let anyone talk you into using a Swiss (Schmidt) Hammer for this task.  It is frought with error when trying to correlate with compressive strength.  Don't be surprised though if someone suggests it (local testing lab or concrete supplier).  In my thinking, they're credibility would immediately suffer (I'm being kind!!).

Drew:

Ron just about has it. There is a lot of literature available on this subject that you should be able to locate. Core sampling is the most common, typically 100mm diameter, ideally diameter at least three times the maximum aggregate size, trimmed to 1:1 length:diameter and crushed in a calibrated machine. The Schmidt Hammer is very sensitive to local surface variations in the concrete. Other near surface tests are Windsor Probe, BRE Internal Fracture Tester and the CAPO and Lok tests used in Scandinavia.